Omega constant

The omega constant ${\displaystyle \Omega }$ is a mathematical constant implied by

${\displaystyle \Omega e^{\Omega }=1}$

with the Euler's number is defined. It applies ${\displaystyle e}$

${\displaystyle \Omega =W(1),}$

where is the Lambert W function . The name comes from the omega function, the second name of Lambert's W function. ${\displaystyle W}$ ${\displaystyle \Omega }$

The first decimal places of are ${\displaystyle \Omega }$

${\displaystyle \Omega =0{,}567143290409783872999968662210\dots }$

properties

• ${\displaystyle \Omega =\ln \left({\frac {1}{\Omega }}\right)}$
• ${\displaystyle \Omega =-\ln(\Omega )}$
• ${\displaystyle \Omega =e^{-\Omega }}$
• If you put on a potency tower that starts with and goes up with , you get : ${\displaystyle e}$ ${\displaystyle -e}$ ${\displaystyle \Omega }$

${\displaystyle \Omega =e^{-e^{-e^{\cdot ^{\cdot ^{\cdot }}}}}}$

• In slightly different words, this means that the limit of the through ${\displaystyle \Omega }$

${\displaystyle \Omega _{n+1}=e^{-\Omega _{n}}}$

is a recursively defined sequence with any start value . ${\displaystyle \Omega _{0}}$

• By

${\displaystyle \Omega =e^{-1}\uparrow \uparrow \infty :=\lim _{n\to \infty }e^{-1}\uparrow \uparrow n}$

the relationship comes in the so-called arrow notation

${\displaystyle \Omega =(1/e)^{(1/e)^{(1/e)^{\cdot ^{\cdot ^{\cdot }}}}}}$

to express that the value of this infinite power tower is with nothing but the same bases , which in turn is only a rather trivial reformulation of the two above properties. ${\displaystyle \Omega }$ ${\displaystyle 1/e}$

• ${\displaystyle \int _{-\infty }^{\infty }{\frac {{\text{d}}t}{(e^{t}-t)^{2}+\pi ^{2}}}={\frac {1}{1+\Omega }}=0{,}638103743365110778522407385519\dots }$
• ${\displaystyle \Omega ={\frac {1}{\pi }}\operatorname {Re} \int _{0}^{\pi }\log \left({\frac {e^{e^{it}}-e^{-it}}{e^{e^{it}}-e^{it}}}\right)\ {\text{ d}}t,}$where the real part of the integral is formed. ${\displaystyle \operatorname {Re} }$
• ${\displaystyle \Omega }$is a transcendent number .

If it were an algebraic number , it would be transcendent according to the Lindemann-Weierstrass theorem . But that contradicts , so that there must be a transcendent number. ${\displaystyle \Omega }$ ${\displaystyle e^{-\Omega }}$ ${\displaystyle e^{-\Omega }=\Omega }$ ${\displaystyle \Omega }$

Individual evidence

1. ↑ Follow A030178 in OEIS
2. ↑ Follow A115287 in OEIS
3. ↑ István Mező: An integral representation for the principal branch of Lambert the W function. Retrieved November 19, 2018 . 

Web links

• Eric W. Weisstein : Omega Constant . In: MathWorld (English).
• Follow A030178 in OEIS